Annulus Area Formula

It is defined as the difference between the area of the two circles of radius R and r. R is the radius of the bigger circle, while r is the radius of the smaller circle. The common area between the two circles is known as the annulus region and the formula used to find that area is known as the annulus formula. It can also be termed as a circle with a hole in the center of radius r. 

Annulus Formula

The shaded area representing the annulus formula is shown below:

The radius of the inner circle = ‘r’

Area of the inner circle = πr²

The radius of the outer circle = ‘R’

Area of the outer circle = πR²

The difference is generally calculated by subtracting a small value from the larger one.  

Annulus formula is given by:

π (R² – r²)

Or

π (R – r)(R + r)

(a² – b² )= (a – b)(a + b))

The area of the annulus region can be calculated if the radius of the inner and outer circle is known.  It is used in various applications, such as finding the area of the circular racing track.  

For example,

If R = 5cm

R = 3cm

Annulus formula will be π (R² – r²)

= π (5 – 3)

= π (25 – 9)

= π (16)

Sample Problems

Question 1: Find the area of the circular racing track formed between the circular field of radius 600m and 300m?

Solution: 

The area of the circular track between the two tracks can be calculated using the annulus formula given by:

π (R² – r²)

Where,

R = 600

r = 300

= π (600² – 300²)

= π (360000 – 90000)

= π (270000)

= 848230 m²

Thus, the area of the circular track is 848230 square meter.  

Question 2: Find the area of the circular racing track formed between the two circles of radius 250m and 400m?

Solution: 

The area of the circular track between the two tracks can be calculated using the annulus formula given by:

π (R² – r²)

Where,

R = 400

r = 250

R is the radius of outer circle and ‘r’ is the radius of inner circle.  

= π (400² – 250²)

= π (160000 – 62500)

= π (97500)

= 306296 m²

Thus, the area of the circular track is 306296 square meter.  

Question 3: Find the area of a path 12cm wide surrounded by a lawn of diameter 300cm?

Solution: 

Given:

Width of the path = 12cm

Diameter of the circular lawn = 300cm

Radius of the inner lawn = diameter/2  

= 300/2

= 150cm

Radius of outer circle = radius of inner circle + width of the path

= 150 + 12

= 162cm

The annulus formula given by:

π (R² – r²)

Where,

R = 162

r = 150

A = π (R – r)(R + r)

A = 3.1415 (162 – 150)(162 + 150)

A = 3.1415 ×12 × 312

A =11761.776 cm²

Thus, the area of the path is 11761.776 square centimeter.  

Question 4: Find the radii of the inner and outer circle if the annulus area and the width of the circular field are 3600m² and 10m?

Solution: 

Given: A = 3600m²

Width = 10m

Width = R – r

Where,

R = radius of outer circle

r = radius of inner circle

R – r = 10

R = 10 + r

The annulus formula is given by:

A = π (R – r)(R + r)

Substituting the value of R in the annulus formula, we get:

A = π (10 + r – r)(10 + r + r)

A = π (10)(10 + 2r)

The area of the width or the annulus area is 3600m². Substituting in the above equation, we get:

3600 = π (10)(10 + 2r)

3600 = π (100 + 20r)

100 + 20r = 1145.9

20r = 1145.9 – 100

20r = 1045.9

r = 52.3m

The radius of outer circle = r + 10

= 52.3 + 10

= 62.3m

Thus, the radii of the inner and outer circle are 52.3m and 62.3m.

Question 5: Find the area of a path 20m wide surrounded by a lawn of diameter 200m?

Solution: 

Given:

Width of the path = 20m

Diameter of the circular lawn = 200m

Radius of the inner lawn = diameter/2  

= 200/2

= 100m

The radius of outer circle = radius of inner circle + width of the path

= 100 + 20

= 120m

The annulus formula given by:

π (R² – r²)

Where,

R = 120

r = 100

A = π (R – r)(R + r)

A = 3.1415 (120 – 100)(120 + 100)

A = 3.1415 ×20 × 220

A =13822 m²

Thus, the area of the path is 13822 square meter.  

Question 6: Find the radii of the inner and outer circle if the annulus area and the width of the circular field are 2400cm² and 15cm?

Solution: 

Given: A = 2400cm²

Width = 15cm

Width = R – r

Where,

R = radius of outer circle

r = radius of inner circle

Difference in the outer and inner radius = width of the field

R – r = 15

R = 15 + r

The annulus formula given by:

A = π (R – r)(R + r)

Substituting the value of R in the annulus formula, we get:

A = π (15 + r – r)(15 + r + r)

A = π (15)(15 + 2r)

The area of the width or the annulus area is 2400cm². Substituting in the above equation, we get:

2400 = π (15)(15 + 2r)

2400 = π (225 + 30r)

225 + 30r = 764

30r = 764 – 225

30r = 539

r = 18cm (approx)

The radius of outer circle = r + 15

= 18 + 15

= 33cm

Thus, the radii of inner and outer circle are 18cm and 33cm.